Lemma 1 
Let \(\rho \in \mathcal {S}(\mathcal {A}\otimes \mathcal {B})\) , let \(\phi _\rho \)  be the map under the linear isomorphism in Eq. (REF ), and let \((\Phi _\rho ,v,\mathcal {K})\)  be a Stinespring dilation of \(\phi _\rho \) , i.e., \(\phi _\rho = v^*\Phi _\rho v\)  with \(v: \mathcal {H}_\mathcal {B}\rightarrow \mathcal {K}\)  linear and \(\Phi _\rho : \mathcal {A}\rightarrow \mathcal {B}(\mathcal {K})\)  a \(C^*\) -homomorphism [1]}. Then \(\phi _{\rho ^{T_\mathcal {A}}} = \phi ^*_\rho = v^*\Phi ^*_\rho v\) .Notably, this is different than so-called co-positive maps, i.e., maps \(\phi ^T := T_\mathcal {B}\circ \phi \) , where \(\phi : \mathcal {A}\rightarrow \mathcal {B}\)  is a completely positive map: by a similar computation we obtain \(\phi _{\rho ^{T_\mathcal {A}}} = \phi ^T_\rho \) , yet, \(\phi ^T_\rho \ne v^*\Phi ^T_\rho v\)  in general.We remark that \(\phi ^* := * \circ \phi \)  denotes the adjoint of the image of the channel \(\phi \) , not its (Heisenberg) dual.

